I am reading Macdonlad's book on "symmetric functions and Hall polynomials" and I have difficulty figuring out an identity which involves hook-lengths. I would like to ask for a hint.

Let $\lambda=(\lambda_1,\dots, \lambda_k)$ be a partition and define $\mu_i=\lambda_i+k-i$, for $1\leq i\leq k$. The hook-length of a box $x\in \lambda$ in the Young diagram is denoted by $h(x)$. One can show the following identity: $$ \sum_{x\in \lambda}t^{h(x)}+\sum_{1\leq i<j\leq k}t^{\mu_i-\mu_j}=\sum_{i=1}^k\sum_{j=1}^{\mu_i}t^j. $$ From this identity, Macdonald concludes the following identity which I am not able to verify: $$ \prod_{x\in \lambda}\left(1-t^{h(x)}\right)\cdot \prod_{1\leq i<j\leq k}\left(1-t^{\mu_i-\mu_j}\right)=\prod_{i=1}^k\prod_{j=1}^{\mu_i}\left(1-t^j\right) $$